Real-rooted atlas

A038324 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(11 + 32\,x + 21\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112132
21214624411024
31331762314553926132768
4146411118043201664074841944811048576
5161051153730558697101120581010696455408410133554432
617715612029242696850215246527820352983015269550666857661211073741824
719487171260419467149149331147456605359059897385103777006416603991317180108854134359738368
821435888132738447282187523522883523625416199317742470304412552136290575617948158495791608378228593611099511627776
923579476914051382850930937832679613781398193643946491300906753421066536695889953922847845541684596374446307673979428004658135184372088832
10259374246014951690151104253951993445216564828757207235234051661016575263463805226369737328781028766986176852020594546922064587370805123910166798809782011125899906842624
11285311670611599154508283157192021245205327554303495265125066188607283033426781318673826381476433565002870201331849773083064672585660155285933709996555201826559836232135027750054222136028797018963968
12313842837672171898540993972754934680436706480412978459722020635921120201695630333590580706321403924815384300442297331516083400722741134195326784952325810832398484201332055294919131864623663007157317273558275113866411152921504606846976
13345227121439318567909468448339814150845677178686990559197402463278818577987602651126721293163012547286801783714221375654753067799987717167839643798634604957831477372582458101563494635724478618663305919050117481798105188333412828966315447237773911946136893488147419103232
143797498335832411014967737031571412594826918619045196178678287636344450493806101009080811927945323856710358255209343364987614803120456749159972977752201217524164954860563256095030755397095262244454347539242681841169703844572697398964809950167278783040512378874617182439699432439199325215086811180591620717411303424
Technical details
Node id
n-4f48af3e25a76d83 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-16a08a122afbb17c

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,          21;
  2:        121,         462,          441;
  3:       1331,        7623,        14553,          9261;
  4:      14641,      111804,       320166,        407484,        194481;
  5:     161051,     1537305,      5869710,      11205810,      10696455,       4084101;
  6:    1771561,    20292426,     96850215,     246527820,     352983015,     269550666,      85766121;
  7:   19487171,   260419467,   1491493311,    4745660535,    9059897385,   10377700641,    6603991317,    1801088541;
  8:  214358881,  3273844728,  21875235228,   83523625416,  199317742470,  304412552136,  290575617948,  158495791608,   37822859361;
  9: 2357947691, 40513828509, 309378326796, 1378139819364, 3946491300906, 7534210665366, 9588995392284, 7845541684596, 3744463076739, 794280046581;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (11+32*x+21*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((11 + 32*x + 21*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 32*x + 21*x^2)*v[i - 1])/(1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.